Completions of quantum coordinate rings

Wang, Linhong

doi:10.1090/S0002-9939-08-09620-2

Completions of quantum coordinate rings
HTML articles powered by AMS MathViewer

by Linhong Wang PDF

Proc. Amer. Math. Soc. 137 (2009), 911-919 Request permission

Abstract:

Given an iterated skew polynomial ring $C[y_1;\tau _1,\delta _1]\ldots [y_n;\tau _n,\delta _n]$ over a complete local ring $C$ with maximal ideal $\mathfrak {m}$, we prove, under suitable assumptions, that the completion at the ideal $\mathfrak {m} + \left \langle y_1,y_2,\ldots ,y_n\right \rangle$ is an iterated skew power series ring. Under further conditions, the completion becomes a local, noetherian, Auslander regular domain. Applicable examples include quantum matrices, quantum symplectic spaces, and quantum Euclidean spaces.

References

V. I. Arnautov, S. T. Glavatsky, and A. V. Mikhalev, Introduction to the theory of topological rings and modules, Monographs and Textbooks in Pure and Applied Mathematics, vol. 197, Marcel Dekker, Inc., New York, 1996. MR 1368852
Michael Artin, William Schelter, and John Tate, Quantum deformations of $\textrm {GL}_n$, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 879–895. MR 1127037, DOI 10.1002/cpa.3160440804
Ken A. Brown and Ken R. Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. MR 1898492, DOI 10.1007/978-3-0348-8205-7
Gérard Cauchon, Effacement des dérivations et spectres premiers des algèbres quantiques, J. Algebra 260 (2003), no. 2, 476–518 (French, with English summary). MR 1967309, DOI 10.1016/S0021-8693(02)00542-2
P. M. Cohn, Skew fields, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, Cambridge, 1995. Theory of general division rings. MR 1349108, DOI 10.1017/CBO9781139087193
K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, 2nd ed., London Mathematical Society Student Texts, vol. 61, Cambridge University Press, Cambridge, 2004. MR 2080008, DOI 10.1017/CBO9780511841699
Karen L. Horton, The prime and primitive spectra of multiparameter quantum symplectic and Euclidean spaces, Comm. Algebra 31 (2003), no. 10, 4713–4743. MR 1998025, DOI 10.1081/AGB-120023129
R. M. Kashaev, The Heisenberg double and the pentagon relation, Algebra i Analiz 8 (1996), no. 4, 63–74; English transl., St. Petersburg Math. J. 8 (1997), no. 4, 585–592. MR 1418255
Tom H. Koornwinder, Special functions and $q$-commuting variables, Special functions, $q$-series and related topics (Toronto, ON, 1995) Fields Inst. Commun., vol. 14, Amer. Math. Soc., Providence, RI, 1997, pp. 131–166. MR 1448685, DOI 10.1016/s0898-1221(96)90020-6
Huishi Li and Freddy van Oystaeyen, Zariskian filtrations, $K$-Monographs in Mathematics, vol. 2, Kluwer Academic Publishers, Dordrecht, 1996. MR 1420862
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Revised edition, Graduate Studies in Mathematics, vol. 30, American Mathematical Society, Providence, RI, 2001. With the cooperation of L. W. Small. MR 1811901, DOI 10.1090/gsm/030
C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 676974
Peter Schneider and Otmar Venjakob, On the codimension of modules over skew power series rings with applications to Iwasawa algebras, J. Pure Appl. Algebra 204 (2006), no. 2, 349–367. MR 2184816, DOI 10.1016/j.jpaa.2005.05.007
Otmar Venjakob, A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory, J. Reine Angew. Math. 559 (2003), 153–191. With an appendix by Denis Vogel. MR 1989649, DOI 10.1515/crll.2003.047
R. Walker, Local rings and normalizing sets of elements, Proc. London Math. Soc. (3) 24 (1972), 27–45. MR 294399, DOI 10.1112/plms/s3-24.1.27